In the spirit of this and this question, I'm interested in the motivation for defining the Euclidean norm in $\mathbb R^n$ to be $\|x\|=\sqrt{\sum_ix_i^2}$. Of course, Euclidean geometry provides a motivation, but the idea is to see how the notion of Euclidean length could arise "naturally" from purely formal considerations of $\mathbb R^n.$
The two linked questions seem to be asking for conceptual motivations, or for a way to formalize the Euclidean postulates within $\mathbb R^n$, if I understand them correctly. My question is different: What are some arguments for viewing the Euclidean norm as the "natural" norm on $\mathbb R^n$, in the sense of exhibiting basic "tameness" properties? In other words, what are some important properties satisfied by the Euclidean norm but not other norms?
I can think of two initial ideas, but both have problems:
Invariance under rotations. The problem here is that "rotations" are defined as length-preserving linear maps, so using this property to define (or even motivate) $\|\cdot\|$ would be circular. One could define rotations as maps whose matrices obey $A^TA=1$, but then one must justify why these maps should be distinguished (without reference to Euclidean norm), which doesn't seem promising.
Only $p$-norm generated by an inner product. I think I've seen a proof of this statement. This is the type of "naturalness" property I'm interested in. But again it is not clear why—without relying on Euclidean geometry—being generated by an inner product is a good property. At least, I think some further insight is warranted here.
Edit: In this post I ask about an idea related to (1) above, which may or may not work out. Basically, the Euclidean norm admits linear isometries besides just reflections along axes, and I wonder if it is the unique norm for which this is true.